Some isolation and stability results for Einstein manifolds

TL;DR

This paper establishes new stability and isolation criteria for various Einstein manifolds, including compact, AH, and ALE types, based on conditions involving the Weyl tensor and other geometric tensors.

Contribution

It introduces novel stability and isolation results for Einstein manifolds under specific tensor conditions, expanding understanding in multiple geometric settings.

Findings

New stability criteria for compact Einstein manifolds.

Isolation results for ALE Einstein manifolds.

Stability conditions for Kähler and Sasaki η-Einstein manifolds.

Abstract

We prove new isolation and stability results for Einstein manifolds in a variety of settings. Imposing conditions on the Weyl tensor, we establish new stability criteria for compact, asymptotically hyperbolic (AH) and asymptotically locally Euclidean (ALE) manifolds and an isolation result in the latter setting. For compact K\"ahler and Sasaki $\eta$-Einstein manifolds, we provide similar results which involve the Bochner tensor and the contact Bochner tensor, respectively.